How to make a fold while following a circle edge?
If I understood the question. I have found 2 answers for it.

Answer 1


Add a circle and go in edit mode and press F to fill
faces.
Add a plane and go in edit mode and press W and subdivide once.
Delete 5 vertices and just have a square shape below.
Select the square and circle and press Ctrl+J (Join).
Go to edit mode and select the corner of the plane and press Ctrl+Shift+B (Bevel) and with the mouse wheel scroll to get segments.




Cleanup Geometry

Go to edit mode and select the verts of the curve and remove doubles and then Press F to fill faces.





Answer 2


Add a plane select to edges and Mean crease in the Properties panel (N) in edit mode.
Add Subsurf modifier view 6.
Add a cloth simulation with default value.





Smooth shading
Add a base cup shape and in physics panel add collision to it.
Play and stop the animation at a point where your satisfied.